The Edgeworth expansion can also be applied to the density of a normalized sum (see e.g. [1]}, [2]}, [3]}, [4]}). Concerning this expansion for the density, we state the suitable density-related functions for our setting. By differentiating eq.tP, for \(r\in \mathbb {N}\)  we get the approximating polynomials for the conditional density function
\(\widetilde{p}_{r,n}(x)=\phi (x)\sum _{*(k_\cdot ,r,\cdot )} H_{r+2u(k_\cdot )}(x) \prod _{l=1}^{r} \frac{1}{k_l!} \Big (\frac{\widetilde{\lambda }_{l+2,n}}{(l+2)!}\Big )^{k_l},\) 
